
being \(\bar{A}=T_{p}A\) , \(\bar{B}=T_{n}B\)  and \(\bar{C}=T_{m}C\) 
as remarked above.
We emphasize that the explicit dependence on the \(p\) 
variable is introduced in () though the
two-periodic embedding in \(\mathbb {Z}^{3}\)  [1]}, [2]}.
In practice this corresponds to the replace \(F^{(\pm )}_{n}F^{(\pm )}_{m}\) 
with \(F^{(\pm )}_{n}F^{(\pm )}_{m}F^{(\pm )}_{p}\)  coherently with the
embedding. Consistency can be checked directly: the value of \(u_{n+1,m+1,p+1}\)  is given by
\(u_{n+1,m+1,p+1} =F^{(+)}_{n}F^{(+)}_{m}F^{(+)}_{p}\frac{N_{[123]}}{D_{[123]}}+F^{(-)}_{n}F^{(+)}_{m}F^{(+)}_{p}\left.\frac{N_{[123]}}{D_{[123]}}\right|_{e_1 \leftrightarrow o_1}\nonumber \\\hphantom{u_{n+1,m+1,p+1} =}{} +F^{(+)}_{n}F^{(-)}_{m}F^{(+)}_{p}\left.\frac{N_{[123]}}{D_{[123]}}\right|_{e_2 \leftrightarrow o_2}+F^{(+)}_{n}F^{(+)}_{m}F^{(-)}_{p}\left.\frac{N_{[123]}}{D_{[123]}}\right|_{e_3 \leftrightarrow o_3}\nonumber \\\hphantom{u_{n+1,m+1,p+1} =}{}+F^{(-)}_{n}F^{(-)}_{m}F^{(+)}_{p}\left.\frac{N_{[123]}}{D_{[123]}}\right|_{{e_1 \leftrightarrow o_1\\e_2 \leftrightarrow o_2}}+F^{(-)}_{n}F^{(+)}_{m}F^{(-)}_{p}\left.\frac{N_{[123]}}{D_{[123]}}\right|_{{e_1 \leftrightarrow o_1\\e_3 \leftrightarrow o_3}}\nonumber \\\hphantom{u_{n+1,m+1,p+1} =}{}+F^{(+)}_{n}F^{(-)}_{m}F^{(-)}_{p}\left.\frac{N_{[123]}}{D_{[123]}}\right|_{{e_2 \leftrightarrow o_2\\e_3 \leftrightarrow o_3}}+F^{(-)}_{n}F^{(-)}_{m}F^{(-)}_{p}\left.\frac{N_{[123]}}{D_{[123]}}\right|_{{e_1 \leftrightarrow o_1\\e_2 \leftrightarrow o_2\\e_3 \leftrightarrow o_3}}, \) 
